The Proximity Rule P: A Minimax Selection Principle on Signed Graphs, with Closed Enumeration on Cycles and a Dichotomy between Complete Graphs and Sparse Meshes (Projective Dynamic Logo Framework — Document D69)

This document studies a single combinatorial object. Given a finite simple graph G and an assignment of states x : V → {±1}, write same(v) for the number of neighbours of v sharing its state, and M(x) = max_v same(v). The proximity rule P selects the assignments minimising M. The rule is invariant under the global inversion x → −x, it is local in the sense that it constrains each vertex separately, and it is minimax rather than aggregate. Four groups of results are established. First, a dichotomy: for connected G, min M = 0 if and only if G is bipartite, and the minimiser is then unique up to inversion; on complete graphs min M = ceil(n/2) − 1, attained at the balanced bipartition, with C(n, n/2)/2 minimisers forming a single orbit. Frustration and degeneracy therefore arrive together, and the imbalance between same-state and opposite-state neighbours is exactly 1 and irreducible whenever the degree is odd. Second, a complete enumeration on cycles. The minimisers of C_n are exactly the states all of whose maximal runs have length one or two; equivalently, the odd-size matchings of C_n. Two closed formulas count them and their Aut-orbits, verified by exhaustive search to n = 15 and extended to n = 31. The total count satisfies T(n) = L(n) + 2 when 3 | n and L(n) − 1 otherwise, with L the Lucas numbers; the period-three correction is ω^n + conj(ω)^n for ω a primitive cube root of unity, so the transfer matrix carries the spectrum {φ, −1/φ, ω, conj(ω)}. Third, a separation of hypotheses. Within the cycle family, orbit multiplicity does not track automorphism strength: |Aut(C_n)| = 2n grows linearly while the orbit count jumps 1, 2, 4, 7, 14, 30. Every odd cycle from C_7 onward has several orbits and every even cycle exactly one, so C_7 is the first member of a family rather than an accident. Fourth, and structurally the most consequential: M is not invariant under switching. Rule P therefore separates configurations that every switching-invariant functional identifies. A companion computation shows that a second layer of states carried by the edges is a pure gauge on complete graphs and carries exactly c − t independent bits on sparse meshes, where c is the cyclomatic number and t the number of independent triangles — so the relation layer is empty on complete graphs and non-empty precisely where triangles are scarce. The rule is not derived from any axiom and resolves no open problem of the PDL corpus. Its status is that of an exploratory hypothesis throughout, and the document is written so that its combinatorial content stands independently of that status. Seven verification scripts are deposited under the same DOI.